Last month, Senate Majority Leader Mitch McConnell unveiled a bill that had been written behind closed doors by a group of Republican senators. McConnell decided not to call a vote on the measure before the Senate's break for the July Fourth holiday as it appeared not to have enough Republican support.

McCain was critical of the process and said the Republicans should "go back to the beginning" and include both parties in the deliberations, letting Democrats have amendments considered to the bill.

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"Introduce a bill," McCain said. "Bring it to the floor. Vote on it. That's the normal process, and if you shut out the adversary or the opposite party, you're going to end up the same way Obamacare did when they rammed it through with 60 votes. Only guess what? We don't have 60 votes."

Democrats have railed against the bill, and groups of moderate and conservative Republicans have criticized it. Over the recess, Republicans have floated various changes to the legislation, which the Congressional Budget Office estimated would lead to 22 million more people uninsured by 2026 compared to Obamacare.